Find rotation matrix to match one 3D vector pair onto another

I have two pairs of 3D vectors named $(A_1, B_1)$ and $(A_2, B_2)$. All four vectors have unit length. I'd like to match one pair onto the other. As I am permitted to assume the angle between $A_1$ and $B_1$ is always the same as the angle between $A_2$ and $B_2$, I imagine that some rotation matrix exists that when applied to both vectors in $(A_1, B_1)$ will give me the same vectors as $(A_2, B_2)$.

Below are some numbers for the sake of a concrete example:

$$A_1 = (1, 0, 0),\quad B_1 = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 0\right),\quad A_2 = (0, 1, 0),\quad B_2 = \left(0, \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right).$$

Note that the angle between $A_1$ and $B_1$ is equal to the angle between $A_2$ and $B_2$ ($45$ degrees).

-
Thanks for the edit, Michael. The question is actually readable now! –  Jono Sep 30 '12 at 2:26

Use procrustes analysis.${}{}{}{}{}{}{}{}$